Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,26)(16,31)(17,35)(18,37)(19,25)(20,23)(21,30)(22,34)(24,32)(27,39)(28,40)(29,41)(33,43)(36,44)(38,42);(5,7,10,6)(8,12,9,11)(13,15)(14,21)(16,32)(17,28)(18,34)(19,30)(20,38)(22,24)(23,40)(25,26)(27,33)(29,36)(31,37)(35,42)(39,41)(43,44);(2,4)(13,16,20,33,19,34,17,29)(14,22,23,41,25,31,35,43)(15,27,28,32,30,36,38,18)(21,39,42,24,26,44,40,37);(5,8,10,9)(6,11,7,12)(13,17,19,20)(14,23,25,35)(15,28,30,38)(16,33,34,29)(18,36,32,27)(21,40,26,42)(22,43,31,41)(24,44,37,39);(5,9,10,8)(6,11,7,12)(13,18,20,27,19,32,17,36)(14,24,23,44,25,37,35,39)(15,29,28,16,30,33,38,34)(21,41,42,31,26,43,40,22)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718631);